Exponents have rules for manipulating them when in ratios. For example:
22/23 = 2*2/2*2*2 = 2(2-3) = 2-1 = 1/2 This is rapid cancellation of factors. It works because the notation it is manipulating is valid.
And for the case when the ratio is equal:
22/22 = 2*2/2*2 = 2(2-2) = 20 = 1
Multiplication is repeated addition, but exponentiation is repeated multiplication. Exponentiation is a method built on a method, not built on units directly as addition is.
Imaginary numbers are a mathematical tool, like a vector or a matrix. A matrix is not something that you measure either - it’s a way of presentation.
Right, but my concern comes from the fact that people think these things *are* physical, actually part of the universe, and that we are simply incapable of perceiving or understanding them. If these concepts were more easily relateable to the real world, or put in the proper context, these confusions wouldn’t occur.
Basically, think to yourself: Suppose you have the equation X^2= -1.
What does this equation mean? Where did it come from? What is it’s source in reality. It has to have one, otherwise I can write 0=1 and say “ok guys, let’s discuss this, it’s a serious problem that needs solution” - obviously one can’t just write whatever one wants - the problem has to come from reality. OK then, so what about the equation above?
The equation only makes sense if you have a vector that can be represented as an arrow on a circle. What kind of entity would require such representation? A voltage that is changing over time, for example. At a given moment in time, in a specific location in a circuit, the voltage has a certain “phase” - a certain shift in time compared to the starting point of measurement. It’s important to keep track of that shift, as well as the amplitude. So one can keep track of the phase by keeping track of where on the circle the “dial” of the voltage is (12 O’clock, 6 O’clock etc’). The way to do it is by using a real axis and an imaginary one; then the voltage can be described by two components:
cos (angle) +/- i sin(angle)
(multiplied by the amplitude).
Then, when you use this form of representation in the equation, you get the result of i or -i which means that the “dial” is on 12 O’clock or 6 O’clock in your circle.
Correct. I dealt with all of that in my circuits courses, and that is definitely the best way to understand imaginary numbers and tie them to reality. Unfortunately, kids are taught about imaginary numbers without this connection to reality being made, and given the impression that the numbers are somehow magical, or that we are simply incapable of grasping their true nature.
Brian, Here’s how I might relate various exponents to reality:
Exponents can be used to describe cell division. Starting with x cells, the number of cells after any number of divisions can be calculated by _ x _ · 2^ _ n _, where n is the number of divisions that have taken place.
Say you have x cells and you want to know how many cells there were n divisions ago. You can calculate this by _ x _ · 2^ _ -n _. So negative integer exponents are instructions for sucessive division in the same way that positive exponents are instructions for successive multiplication. In order for the process to be continuous and mathematically sensible, you must define _ x _ ^0 = 1 (the multiplicative identity). This too can be related to reality though: how many cells are there after 0 divisions (i.e. now)… x · 2^0 = x · 1 = x.
Now, say you didn’t know that the number of cells doubled with each division. The only information you have is that you started with w cells and ended with x cells after n divisions. The cellular multiplication per division can be calculated by (_ x _ / _ w )^(1/ n _).
That covers positive, zero, negative, and (simple) rational exponents. As one’s math training advances you can discuss formalities/identities like a^(m+n) = a^m · a^n and more complex concepts like irrational exponents (these are tough). It’s been a while since I studied it, but I believe irrational exponents are only truly calculable through series expansion.
Does that help?
That covers positive, zero, negative, and (simple) rational exponents. As one’s math training advances you can discuss formalities/identities like a^(m+n) = a^m · a^n and more complex concepts like irrational exponents (these are tough). It’s been a while since I studied it, but I believe irrational exponents are only truly calculable through series expansion.
Does that help?
Or as the limit of a sequence. Let the sequence {a\sub n} converge to a, where a\sub n is rational. Then we can write b^a = lim (n -> inf) b^a\sub n. Since each a\sub n can be written in the form M/N (we assume a\sub n is rational) the b^(M/N) is just the N-th root of b^M a well defined quantity. The inverse to exponentiation is the logarithm which can be defined by a convergent series for any real positive value. [We are assuming the system of real numbers. For complex numbers this definition can be extended]
Bob Kolker
Or even more simple, take exponents. 2^2 apples is 2 sets of 2 apples. Well, that’s 4 apples. 1^3 apples is one set of (one set of one apple) which is just one apple. But now what about 2^0? Zero sets of two apples? Well, that is one apple. And 2^-1? A negative set of two apples? Well, that is half an apple.
You’re fostering the same problem you’re trying to correct. Explain to a kid why zero sets of one apple is not zero, or how many sets of sets 2^(3/2) equals. I don’t think traditional methods of teaching imaginary and negative numbers have any less relation to reality than yours do.
The important similarity between your method and traditional methods is that they are correct within certain situations – they are contextual, and that’s not exactly a problem that we as philosophers can (or would even want to) solve.
Hmm, I just need to clarify my point. Your example is true, but that’s not what I meant. For instance, a one dollar bill physically exists. It is an entity. And there is one of those bills. But you cannot have -1 of those bills. Do you understand what I mean? The least quantity any entity can be is 1. If an entity has 0 quantity, then it does not exist. Therefore, in this sense, negative quantities are an impossibility and would imply some alternative to existence vs. non-existence, like…less than non-existence.
Even when your liabilities exceed your assets, the fact that your net worth is a negative number is not an actual entity. This is still a concept of method. Physically what is going on is you lose all your assets, but you don’t continue losing assets once you go negative. You have no more assets to lose when you have 0 assets! No, what happens is you go in debt. You don’t have anymore assets, so you are going to have to get more in the future. I hope my point is clear.
Your point is clear, but it does not support a conclusion that negative numbers are somehow invalid. I suggest you read what has been written in this thread about the validity of negative numbers and understand why they are not only perfectly valid but also absolutely necessary. What you have written above seems to suggest that only the counting numbers (i.e., 1, 2, 3, 4, 5, …) are valid. Maybe you would also accept rational numbers formed from the counting numbers? If we (humankind) were to accept only counting numbers and rational numbers formed from the counting numbers we would set science and technology backwards many centuries!
John Link
Here’s how I think of exponents.
Addition is adding numbers together. Easy to validate to sense perception.
2 + 4 = 6.
Multiplication is a “step up” from addition. Mutliplying a number is just adding it to itself a bunch of times.
2 * 4 = 2 + 2 + 2 + 2 = 8
I think of exponents as a “step up” from that.
24 = 2 * 2 * 2 * 2 = 16
Are you seriously saying that they didn’t teach it to students this way? How old are you? Schools must be even worse now than when I was in school. I had a huge problem with square roots until I finally realized that they were just the opposite of squaring, much like division is the opposite of multiplication. It wasn’t explained to me. I had to eventually figure it out on my own.
I hate math. I honestly hate it. I have ever since high school. I only know what I have to know to be a practical programmer.
Yet in elementary school, I was so good at it, me and three or four other students in my school got to start learning basic algebra before everyone else. Once I got to high school, I’m not certain how, but it became different. It was no longer interesting or fun as it became increasingly, nonsensically complex.
If you’ve been paying attention to the 1 = .9999… repeating thread, you know that I take huge issue with that. That the entirety of the mathematical world accepts such nonsense only adds to my distaste for math as a subject.
Obviously there is no such quantity as -1 in real life, ..
Neither is there a quantity as +1 in real life. Things like 1 cow or 1 dollar do exist in real life, but not the number 1. The latter is an abstraction that can be useful in counting real things, so that an abstract concept like natural number has a useful application in real life. Exactly the same holds for other abstract notions like negative numbers, rational numbers, real numbers or complex numbers. These can all be used for real-life applications, in the case of negative numbers for example for bookkeeping (an extension of the notion of counting) and in the case of complex numbers for example for electrical circuits. In general the connection between the abstract concept of a certain kind of number will be more “complex” (no pun intended) for more general categories, like complex numbers, than for natural numbers (counting things). That in the latter case the link between number and real object is simple (counting) does not mean that the number (an abstract concept) and the object (cow, dollar, real objects) are the same. So when you realize that all numbers are abstract and there doesn’t exist any number in reality, but that those numbers and the corresponding theories can be applied to real life situations, there isn’t any problem.
If you’ve been paying attention to the 1 = .9999… repeating thread, you know that I take huge issue with that. That the entirety of the mathematical world accepts such nonsense only adds to my distaste for math as a subject.
I haven’t followed that thread, but the fact that 1 = .9999…repeating follows by elementary logic from the mathematical definitions and notation, it’s no nonsense, on the contrary, it’s the only logical answer. But perhaps you don’t like logic…
I’ll try not to spark that debate in this thread. I don’t understand high-level concepts in math, but thus far in my life, mathematical concepts that I don’t understand have only given me frustration, I imagine due to how badly they were taught.
I couldn’t understand most of the proofs I’ve seen for 1 = .999… But I haven’t understood a proof yet that I didn’t see a fallacy in.
I’m going to be calm in this topic, so there’s no need for personal insult.
Back to the topic, although, we were talking about mathematical concepts that have no relation to reality anyway…
Yet in elementary school, I was so good at it, me and three or four other students in my school got to start learning basic algebra before everyone else. Once I got to high school, I’m not certain how, but it became different. It was no longer interesting or fun as it became increasingly, nonsensically complex.
If you’ve been paying attention to the 1 = .9999… repeating thread, you know that I take huge issue with that. That the entirety of the mathematical world accepts such nonsense only adds to my distaste for math as a subject.
Such nonsensical complexity is required to do physics. No nonsensically complex mathematics, no physics.
I sense that you are not comfortable with the abstract nature of mathematics. Hey, forget Modern. Look at Isaac Newton’s analysis of the tides. Or, maybe not. It might cause your head to explode. Think about it. Mathematical operations on quantities smaller than you can imagine and yet they are not zero (infinitesimals). You are on the same page as Bishop Berkeley who gave a rather sophisticated refutation of the notion of infinitesimal (see -The Analyst- by George Berkeley). He was so on point that embarrassed mathematicians had to invent the concept of the limit (the major accomplishment of 19th century mathematics). Infinitesimals did not make a respectable comeback until 1960 with the work of Abraham Robinson on hyper-reals and non-Archimedian fields.
It is all nonsensical complexity, which is why you have a computer to complain on and with.
Bob Kolker
Right, but my concern comes from the fact that people think these things *are* physical, actually part of the universe, and that we are simply incapable of perceiving or understanding them. If these concepts were more easily relateable to the real world, or put in the proper context, these confusions wouldn’t occur.
The imaginary number i is the abstract mathematical idealization of a 90 degree (pi/2) counter clockwise turn of a unit length rigid rod. In a word, the imaginary unit is a rotation which is not all that removed from the real world. A proper understanding of complex numbers goes back at least 150 years. Even further if you include Euler’s remarkable formula (I consider this the most beautiful equation in mathematics).
Behold!
exp (i*pi) = -1.
Thinking about it gives me goose bumps and makes the hair on the back of my neck stand up.
Bob Kolker
I hate math. I honestly hate it. I have ever since high school. I only know what I have to know to be a practical programmer.
Do you know what a computable functions is? As a practical programmer you should.
You are going to love the Ackermann Function which is computable and not all that difficult to program using any of the standard high level languages:
Here it is:
A(0, n) = n + 1
A(m, 0) = A(m-1, 1) if m > 0
A(m, n) = A((m-1, A(m, n-1)) if m > 0 and n > 0
This simple looking function encapsulates succession (adding 1), addition, multiplication, exponentiation, superexponentiation and an infinite (unllmited) sequence of operations beyond that. Super-exponentiation is the application of exponentiation repeatedly. For example,
a^(a^(a^(…))) n times.
the value for A(4,2) has nearly 20,000 digits in the decimal system and anyone who claims to have computed anything beyond A(4,4) is probably a fibber.
Since you are a practical programmer you might want to fiddle with this function. It can be programmed by by iteration and recursion. This function is used as an exercise in computer science courses.
For details see
http://en.wikipedia.org/wiki/Ackermann_function
Bob Kolker
Back to the topic, although, we were talking about mathematical concepts that have no relation to reality anyway…
Almost every mathematical subject written on in the journals has a connection to physical reality. Sometimes the connection is long and abstract consisting of abstraction of abstraction (for example Category Theory). However all mathematics connects historically to answering two questions:
How many?
How much?
The most abstruse differential equation or topological theorem can be traced back to these questions, sometimes along a long and difficult path.
Some mathematics connects to visual arts, such a group theory which connects to symmetry, and projection. Any picture with a vanishing point has a connection to Projective Geometry, which is the geometry of incidence, rather than distance.
Bob Kolker
Here’s how I think of exponents.
Addition is adding numbers together. Easy to validate to sense perception.
2 + 4 = 6.
Multiplication is a “step up” from addition. Mutliplying a number is just adding it to itself a bunch of times.
2 * 4 = 2 + 2 + 2 + 2 = 8
I think of exponents as a “step up” from that.
24 = 2 * 2 * 2 * 2 = 16
Are you seriously saying that they didn’t teach it to students this way?
Please go back and re-read the discussion, because you’ve completely skipped over my point. The examples you mention are perfectly fine - and they’re exactly what I’ve said multiple times now (which you’ve clearly skipped over) -, but it all breaks down as soon as the exponent drops below 1, or goes negative, so I’m saying it needs to be explained to kids in a better way that takes the full context of the possibilities of exponents into account from the beginning.
How old are you?
Three years older than you, with a degree in physics. Great job confusing the intent of the discussion. Read the rest of your *own* post and you’ll understand the true intent. People come to hate math - and I came to hate QM - because it is not explained in a proper conceptual way. Finding that proper conceptualization is the point of this discussion.
Brian,
Try this link, Pisaturo has a degree in math from MIT and is an Objectivist:
Ah, you’re right. I did kinda miss that point. I don’t know how to deal with exponents below 1, so I couldn’t say anything about that.
I asked how old you were, because if you were younger, it would mean that you started school later than me, and that I could assume schools have gotten worse. I meant no insult.
From Pisaturo
1 Introduction
In Introduction to Objectivist Epistemology (ITOE), Ayn Rand writes:
Since axiomatic concepts are not formed by differentiating one group of existents from others, but represent an integration of all existents, they have no Conceptual Common Denominator with anything else. They have no contraries, no alternatives. The contrary of the concept “table”—a non-table—is every other kind of existents. The contrary of the concept “man”—a non-man—is every other kind of existents. “Existence,” “identity” and “consciousness” have no contraries—only a void.[1]
In this essay, I will argue that the axiomatic concept of “existence” does indeed have Conceptual Common Denominators (CCDs) and, not literally “contraries,” but rather what I will call complements. I will also show that identifying the CCDs and complements sheds important light on this axiomatic concept. (An investigation into the CCDs and complements of the axiomatic concepts of consciousness and identity is outside the scope of this essay, except to the extent that it relates to the present analysis of “existence.”)[2]
Hmm…
[Edit: I’m removing my previous post–it was written in regard to Ifat, and I somehow didn’t see the long list of people who posted after her.]